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GATE 2026 ST – Question 52

Probability · Moments and Inequalities · 2 marks · Multiple select

X has full support {1,2,3,…}, E(X²)<∞ and CDF F. Which hold?

  1. E(X)=Σₙ₌₁∞[1−F(n−1)]
  2. F(X) is discrete uniform
  3. Some such X has the same distribution as 1/X
  4. E(X²)=Σₙ₌₁∞(2n−1)[1−F(n−1)]

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Show answer and explanation

Correct answer: (A) E(X)=Σₙ₌₁∞[1−F(n−1)]; (D) E(X²)=Σₙ₌₁∞(2n−1)[1−F(n−1)]

Explanation

For positive integer X, write X=Σₙ₌₁^X1 and X²=Σₙ₌₁^X(2n−1). Taking expectations via nonnegative sums gives A,D. The transform F(X) has generally unequal point masses, so B fails. Full support on every positive integer gives positive probability X>1, whereas 1/X≤1; hence C fails. If “support” meant only that X takes values in this set, the degenerate X=1 would change C; the official **A,D** uses full support.

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